Non-Synchronous Orbital Skyhooks for the Moon and Mars
with Conventional Materials
Hans Moravec 1978
Abstract: A satellite in low circular orbit has two huge tapered
cables extending outwards and rotating in the orbital plane, touching the
planet each rotation. The tip velocity cancels the orbital velocity at the
contacts, as in a rolling wheel. It can gently lift loads from the
surface and accelerate them to escape velocity, and capture and lower
speeding masses. Taper is minimized when the satellite's radius is one
third the planet's, and for Mars and the moon is reasonable with existing
materials such as fiberglass and Kevlar.
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The idea of a planet to orbit transportation system involving
an enormous tapered cable extending from a synchronous satellite to
the ground has been in the literature for almost two decades [1, 2,
3]. It has hitherto been considered applicable only in the distant
future, when materials stronger than any now available come into
existence.
This report points out that the combination of a new material,
Kevlar [4] and a new, less expensive, satellite skyhook configuration
[5, 6] makes skyhook transportation feasible now on bodies as large as
Mars. On the moon, in particular, a Kevlar skyhook has enormous
advantages over rockets for the supply and crew rotation missions
envisioned for space industrialization efforts [7].
A synchronous skyhook is made by lowering a cable from a
synchronous satellite to the surface, balanced by an even longer cable
extending outwards from synchronous orbit. Anchored to the ground and
put into tension by a ballast at its far end, it would be a cosmic
elevator cable, able to deliver mass to high orbit with extreme
efficiency, also providing a means for extracting the rotational
energy of the planet. Such a structure cannot reasonably be built on
Earth given existing structural materials. It would be possible if a
cable with 10 times the strength/weight ratio of steel, or 1/8 the
theoretical strength/weight ratio of crystalline graphite could be
fabricated. A graphite cable with a density of 2.2 g/cm^3 and a
tensile strength of 2.1e11 dyne/cm^2 could be fashioned into a
synchronous terrestrial skyhook which had only 100 times ground level
cross section at synchronous orbital height. At any one time it could
support one powered elevator massing 1/6000 of the cable mass [6].
Mars has a much shallower gravity well, and a synchronous
skyhook for it is almost reasonable with conventional
materials. Kevlar is a new superstrong synthetic from the DuPont Co.
With a density of 1.44 g/cm^3 and a tensile strength of 2.76e10
dyne/cm^2 it has about 5 times the strength/weight of steel. Stressed
to half this, to build in a safety factor of two, Kevlar can be used
to construct a synchronous martian skyhook with a taper of 16,000:1,
able to support one millionth of its own weight at a time. The numbers
for the moon, which has little gravity, but rotates very slowly, are
17.5:1 and 1/13,000.
Figure 1: A non-synchronous skyhook's progress around a planet: two spokes
of a giant wheel.
[6] introduced the concept of a non-synchronous skyhook.
Figure 1 illustrates the idea. A satellite in low circular orbit is
elongated enough to just touch the surface in certain positions. It
spins so that, like a rolling wheel, its rotation cancels its
tangential velocity during the contacts with the surface. Such a
structure can be constructed to orbit at any height, and a synchronous
skyhook is a special case.
In very high orbits the forces on the cable must be integrated
over long distances, resulting in large tapers. For very low orbits,
the satellite must spin rapidly to keep the contact points stationary,
and the quadratic dependence of centrifugal force on rate of spin
results in a large taper in the limit. The taper is minimized between
these extremes, when the radius of the skyhook is about 1/3 the radius
of the planet.
An optimum size skyhook of this kind touches down six times
per orbit. It is much smaller than the synchronous variety for the
earth, moon and Mars, but its length is still enormous by conventional
standards. Because of its scale, its motion near the ground during a
touchdown is purely vertical. It appears to descend with a constant
upward acceleration, coming to a gentle momentary stop, then ascending
again. This acceleration is 1.4 gravities on Earth, 0.28 g on the
moon and 0.5 g on Mars.
A load attached to the bottom end of such a skyhook during a
touchdown will be accelerated to a maximum of 1.6 times escape
velocity at the highest point of the cable end's trajectory.
Launching a mass in this manner extracts rotational and orbital energy
from the skyhook, and lowers the orbit. Conversely, a high velocity
craft which rendezvous with and attaches itself to the upper end of
the cable, and is then decelerated and lowered to the ground, injects
a similar amount of energy. Simultaneous docking of equal masses at
both ends of a skyhook would leave the orbit essentially unchanged.
The most plausible way to operate a device like this may be to have
the cable ends merely approach the surface at a safe distance. A
small rocket could be used to match the relatively tiny velocity and
position differences between the cable tip and the ground. It would
then be possible to borrow and deposit small amounts of orbital energy
without risking collisions of the cable and surface.
TABLE I. Parameters for Optimally Sized Skyhooks
Orbital Liftoff Fiberglass Kevlar
Body Period(hr.)Accel(g) Taper Mass Taper Mass
Mercury 2.37 0.57 2200 23000 49 350
Venus 2.37 1.39 1.2e^20 3.0e^21 1.3e10 2.3e11
Earth 2.16 1.40 7.2e21 1.9e23 1.0e11 1.9e12
Moon 2.78 0.28 13 72 3.6 13
Mars 2.62 0.49 17000 200000 136 1100
Ganymede 3.41 0.26 35 240 6.0 28
Titan 3.39 0.26 29 190 5.4 24
Table I lists parameters for optimum size fiberglass and
Kevlar skyhooks for some solar system bodies. Fiberglass is assumed to
have a density of 2.5 g/cm^3 and a tensile strength of 2.41e10
dyne/cm^2. Kevlar has a density of 1.44 g/cm^3 and a tensile strength
of 2.76e10 dyne/cm^2. Orbital period is how long it takes the skyhook
to make a full circuit of the body. The liftoff acceleration is the
vertical acceleration experienced by a skyhook payload near the
ground, not including the surface gravity of the planet. It gives an
indication of how long the touchdown lasts. Taper is the ratio in
cross sectional area between the center of the skyhook, where it is
thickest, and the tips, where it is thinnest. The Mass columns give
the ratio between the mass of the skyhook and the largest payload that
it can support at one time at each end. Thus a lunar Kevlar skyhook
can lift 1/13 of its own mass. The numbers assume the skyhooks are
stressed to at most half the tensile strength of the material of which
they are made, thus incorporating a safety factor of two.
Evidently Earth and Venus are too large for Kevlar skyhooks.
Kevlar is strong enough for Mars, Mercury and all the moons of the
solar system.
Some current plans for space industrialization call for
transport of large quantities of equipment and people to and from the
moon. The proposed linear accelerator mass driver [7] is ideal for
launching ore from the moon. It provides no way of bringing payloads
down to the surface, and with its 1000 g accelerations and small mass
unit is unsuitable for launching bulkier and more delicate loads.
A Kevlar lunar skyhook is able to lift and deposit 1/13 of its
own mass every 20 minutes, and subjects payloads to a maximum 0.45 g
of acceleration. It would seem to be a desirable alternative to
expensively fuelled rockets for routine supply and crew rotation
missions to the moon's surface.
The tapers for non-synchronous skyhooks used in this report
were obtained by integrating the forces on the cable between ground
level and satellite center, at the instant of a touchdown. This is
when the stress is at its highest [8].
Define
r[p] the radius of the planet
m[p] the mass of the planet
w[p] the rotation rate of the planet (in radians per unit time)
r[o] the radius of the orbit
w[o] the orbital rate of the satellite's mass center
w[s] the rotation rate of the satellite
d the density of the cable material
t the tensile strength of the cable material
A(r) cable cross section at distance r from the planet center
G the universal gravitational constant
to make contact point stationary,
w[s] = (r[o] w[o] - r[p] w[p]) / (r[o] - r[p]))
and for a circular orbit
w[o] = sqrt(G m[p] / r[o]^3)
this last substitution is only an approximation, since the extended
satellite does not orbit and rotate exactly like a point at its mass
center.
The stresses in the cables are caused by their weight in the
planet's gravitational field and the accelerations due to the orbital
motion and spin of satellite. They are maximum in the downward hanging
cable. Both cables must be built to take this stress and the satellite
is thus symmetric about its center. If the cables are constructed so
as to make the tension per unit area constant, the cross section of
the downward hanging cable at distance r from the planet center is
given by
A(r) = A(r[p]) exp( d/t (r - r[p]) ( G m[p]/(r r[p]) - r[o] w[o]^2 +
(r[o] - (r + r[p])/2) w[s]^2 ) )
The mass ratios were found by numerically integrating this
expression over r. Some confidence in the general stability of
skyhooks of this kind has been obtained by observing computer
simulations of optimum size terrestrial graphite versions [6]. The
only serious problems revealed were caused by launches not
complemented by captures. These lowered the satellite's orbit and
caused collisions with the ground.
References and Notes
[1] Y. Artsutanov, Komsomolskaya Pravda, July 31, 1960 (contents
described in Lvov, Science 158 946 (1967)).
[2] J.D. Isaacs, A.C. Vine, H. Bradner, G.E. Bachus, Science 151 682
(1966) and 152 800 (1966) and 158 946 (1967).
[3] J. Pearson, Acta Astronautica 2 785 (1975).
[4] J.H. Ross, Astronautics & Aeronautics, 15-12 44 (1977).
[5] The central idea in this paper, of a satellite that rolls like a
wheel, was originated and suggested to me by John McCarthy of
Stanford.
[6] H.P. Moravec, Advances in the Astronautical Sciences, 1977, also
J. Astronautical Sciences 25 (1977).
[7] G.K. O'Neill, The High Frontier, Human Colonies in Space (William
Morrow & Co., New York, 1976).
[8] The derivations were done using the MACSYMA symbolic mathematics
computer system at MIT.